Sometimes we wash with a cup, sometimes not; sometimes we wash twice, sometimes thrice; and not everyone agrees on method or pattern in all cases. Who says which ways we are supposed to wash, in which situations, and why?

3 Answers
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There are, of course, two reasons to wash hands. One is physical cleanliness the other is for Taharah. I think physical cleanliness is understood. Hands need to be completely physically clean before washing them for Taharah.

Originally, DeOraisah, there was no specific Tumah of the hands. If something made a person Tamei then their whole body was Tamei. The Sanhedrin decreed that hands could become Tamei derabbanan under certain circumstances. The first such decrees came in the time of King Solomon and were expanded by the Sanhedrin over time. These decrees were made so that people should be careful about touching certain things such as Torah scrolls or other holy writings or Terumah. Later this was expanded even to touching Chullin (as a precaution for Terumah) so that today we need to wash before eating bread. Tumah of the hands also occurs if we touch certain parts of our body that are normally covered. It is possible that while a person sleeps this may happen or if a person is not paying attention this may happen so that we need to wash hands before doing certain things such as saying a berachah or praying or learning Torah. There is also an additional decree that water that touches hands that are Tamei can make other things Tamei. That is Tamei hands are Tamei in the second degree but water that touches them becomes Tamei derabbanan in the first degree. Today we only continue to do this as a takanah since everyone has a status of second degree Tumah out of doubt anyway.

For washing the hands to be effective in terms of Taharah it must be done in one of two ways. One way is to put your (physically clean) hands once into a valid mikvah at least up to the wrists. This can be done with both at once or with either hand first so long as the Tamei hand doesn't touch the Tahor hand in between. For example, if you dipped your right hand into the Mikvah, took it out and then touched it with the still Tamei left hand then the right hand would become Tamei again so it would have to be dipped again.

The other way is to pour water water from a vessel over them. The vessel has to be a complete unbroken vessel of at least a certain size. Each hand needs to be washed at least twice. Once to purify the hand and then a second time to purify the water on the hands (because of the decree that such water becomes Tamei). The first washing must be exactly up to the wrist. If it doesn't go up to the wrist it doesn't purify the hand. If the water from the first washing touches any part of the body above the wrist it makes the body (including the hand) Tamei again. If the Tahor hand is touched by a Tamei hand or by Tamei water then it becomes Tamei again.

To prevent these problems and make sure the hands are properly purified various customs and practices of hand washing were instituted. Some people also have the custom to wash even more than two times.

In practice today we wash before eating bread or upon waking up from a sleep of at least half an hour. One should also wash hands before learning Torah, praying or saying a berachah unless one has had his attention on his hands to make sure they have remained clean.

You are assuming that washing after touching unclean parts of the body prior to tefilah requires a keli (it sounds like you want to apply this to removing ruach ra'ah as well). Do you have a source for that?
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YDKNov 7 '11 at 3:00

What I said was that for washing to be effective in terms of Taharah it needs to be either in a valid mikvah or poured from a complete unbroken vessel. This is clear from the Mishnah that I referenced above. The law about a vessel is in Yadayim 1:2.
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follickNov 9 '11 at 3:25

"Tumah of the hands also occurs if we touch certain parts of our body that are normally covered. It is possible that while a person sleeps this may happen or if a person is not paying attention this may happen so that we need to wash hands before doing certain things such as saying a berachah or praying or learning Torah." yadayim askanios would require a cohen to wash with a keli for teruma (although touching his body would not since the rest of his body is obviously tahor, otherwise his entire body would need a mikvah). It would not require a keli for nekius for davening.
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YDKNov 9 '11 at 4:45

You mention that one needs to wash twice per hand for tahara to remove the tamei water. The shulchan aruch and rama in OC 162:2 paskin that this is only if he didn't use a full reviit the first time. But, if he used a full reviit the first time, there is no tamei water and there is no need to wash a second time. Most people nowadays I imagine use at least a reviit of water per time. In fact, I have tried once not to and found it exceedingly difficult to cover the whole hand. In short, we almost never require 2 washes per hand.
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Double AA♦Dec 20 '11 at 17:28

Maran writes (Shulhan Arukh 4:1) that one shall wash his hands in the morning and make the Berakha "Al Netilat Yadayim." The Rama writes (ibid.) that one that did not use the restroom recites the Berakha of "Asher Yasar" anyways. However, we do not follow this Pesaq, because Rabenu HaAri writes (Sha'ar HaKawanot pg. 2) that one does not make a Berakha if he did not use the restroom (and the Rashash also writes so). Maran continues (ibid.) that waters that are unusable for washing for a meal are permitted for Netilat Yadayim in the morning, however some one doesn't bless upon them.

The Halakha Berura brings down five opinions as to the reason one washes their hands in the morning (however the answer of the Ari HaQadosh is not written in Halakha Berura): Introduction- The Gemara writes (Berakhot 60b) that one makes the Berakha "Barukh Ata A-donai E-lohenu Melekh HaOlam Asher Qideshanu Miswotaw WeSiwanu Al Netilat Yadayim."

The answers:

The first answer is in the Zohar HaQadosh (Parashat WaYesheb 184b) when one sleeps his pure soul leaves his body and he tastes the taste of death and the evil spirit rests upon his body and even when his soul returns the evil spirit is still present, and in order to remove this one must wash their hands

The second answer is from the Gemara in Shabbat (108b-109a) the evil spirit rests upon one's hands at night and it doesn't leave until one washes their hands properly. This has nothing to do with the sleeping, it has to do with the night.

The third is from the Rosh (Berakhot 9:23) one's hands are disgusting because while one sleeps their are bound to touch disgusting places.

The fourth answer is from HaRashba HaQadosh (Shu"t 1:191) in the morning a person is as a "new creation" as it written (Ekha 3:23) "New to the morning; Your faithfulness is great." For this reason we are required to thank Hashem that He created us for His honor and to respect His name.

The fifth answer is from Rabbi Dawid Abuderham (Hilkhot Netilat Yadayim Shaharit) just like the Kohanim must wash their hands before the service of the Hashem in the Bet HaMiqadash so too a man must wash his hands prior to serving his Creator, in order to serve Him with clean hands.

However the sixth answer is brought down in the Sefer of the Ari HaQadosh "in whose words we live by [a saying by Rabenu Masliah Ma'azuz]" Sha'ar HaKawanot (pg. 2) one's hands touch disgusting places and therefore one must wash them (so far- same as the Rosh). One must wash their hands in order to remove the Qelipah referred to as "Shibta" and that is the reason that we wash our hands in the morning. It is very important to note: it DOES make a practical difference in reason for the washing because according to some one would be unable to bless in some situations (Halakha Berura ibid.).

Washing for tefillo and when getting up are mentioned together.
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Avrohom YitzchokNov 7 '11 at 17:48

That's because they are related. When you get up, you must wash your hands for tefila either because they are unclean or because of kedusha, neither which require 3x. The status for which you need 3x is ruach raa which is not discussed in that s"k.
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YDKNov 8 '11 at 4:08